This achievement can only be unlocked after completing 'Weathertop' in story mode, which will unlock Aragorn as a playable character.
The Mushroom Crown is a treasure item you'll find while playing free play on the second chapter of the story mode, 'The Black Rider'. It is located shortly before the level ends, in a pile of dirt that Sam or Tom Bombadil can dig up with their spade.
Once you have the item, press and hold . From here you can select ANY character. Be sure to select just plain old Aragorn, and not any of his variants, as those won't work for the achievement. Then, press and hold and select fifth circle going clockwise. Find the Mushroom Crown among your many treasures, equip it, and the achievement will unlock.
If you wish to unlock this achievement in Middle Earth rather than in a story chapter in free play, you will need to wait until you've completed all of the story chapters, since free play selection of characters is not available in Middle Earth until then. Aragorn may also be available for you depending on your progress in the story.